| add.trig {bsts} | R Documentation |
Trigonometric Seasonal State Component
Description
Add a trigonometric seasonal model to a state specification.
Usage
AddTrig(
state.specification = NULL,
y,
period,
frequencies,
sigma.prior = NULL,
initial.state.prior = NULL,
sdy = sd(y, na.rm = TRUE),
method = c("harmonic", "direct"))
Arguments
state.specification |
A list of state components that you wish to add to. If omitted, an empty list will be assumed. |
y |
The time series to be modeled, as a numeric vector. |
period |
A positive scalar giving the number of time steps required for the longest cycle to repeat. |
frequencies |
A vector of positive real numbers giving the number of times each cyclic component repeats in a period. One sine and one cosine term will be added for each frequency. |
sigma.prior |
An object created by |
initial.state.prior |
An object created using
|
sdy |
The standard deviation of the series to be modeled. This
will be ignored if |
method |
The method of including the sinusoids. The "harmonic" method is strongly preferred, with "direct" offered mainly for teaching purposes. |
Details
Harmonic Method
Each frequency lambda_j = 2\pi j / S where S
is the period (number of time points in a full cycle) is associated
with two time-varying random components: \gamma_{jt}, and gamma^*_{jt}. They evolve through
time as
%
\gamma_{j, t + 1} = \gamma_{jt} \cos(\lambda_j) + \gamma^*_{j, t} %
\sin(\lambda_j) + \epsilon_{0t}
%
\gamma^*_{j, t + 1} = \gamma^*[j, t] \cos(\lambda_j) - \gamma_{jt} *
\sin(\lambda_j) + \epsilon_1
where \epsilon_0 and \epsilon_1 are
independent with the same variance. This is the real-valued version
of a harmonic function: \gamma \exp(i\theta).
The transition matrix multiplies the function by
\exp(i \lambda_j, so that
after 't' steps the harmonic's value is
\gamma \exp(i \lambda_j t).
The model dynamics allows gamma to drift over time in a random walk.
The state of the model is
(\gamma_{jt}, \gamma^*_{jt}),
for j = 1, ... number of frequencies.
The state transition matrix is a block diagonal matrix, where block 'j' is
\cos(\lambda_j) \sin(\lambda_j)
-\sin(\lambda_j) \cos(\lambda_j)
The error variance matrix is sigma^2 * I. There is a common sigma^2 parameter shared by all frequencies.
The model is full rank, so the state error expander matrix R_t is the identity.
The observation_matrix is (1, 0, 1, 0, ...), where the 1's pick out the 'real' part of the state contributions.
Direct Method
Under the 'direct' method the trig component adds a collection of sine and cosine terms with randomly varying coefficients to the state model. The coefficients are the states, while the sine and cosine values are part of the "observation matrix".
This state component adds the sum of its terms to the observation equation.
y_t = \sum_j \beta_{jt} sin(f_j t) + \gamma_{jt} cos(f_j t)
The evolution equation is that each of the sinusoid coefficients follows a random walk with standard deviation sigma[j].
\beta_{jt} = \beta_{jt-1} + N(0, sigma_{sj}^2)
\gamma_{jt} = \gamma_{j-1} + N(0, sigma_{cj}^2)
The direct method is generally inferior to the harmonic method. It may be removed in the future.
Value
Returns a list with the elements necessary to specify a seasonal state model.
Author(s)
Steven L. Scott steve.the.bayesian@gmail.com
References
Harvey (1990), "Forecasting, structural time series, and the Kalman filter", Cambridge University Press.
Durbin and Koopman (2001), "Time series analysis by state space methods", Oxford University Press.
See Also
Examples
data(AirPassengers)
y <- log(AirPassengers)
ss <- AddLocalLinearTrend(list(), y)
ss <- AddTrig(ss, y, period = 12, frequencies = 1:3)
model <- bsts(y, state.specification = ss, niter = 200)
plot(model)
## The "harmonic" method is much more stable than the "direct" method.
ss <- AddLocalLinearTrend(list(), y)
ss <- AddTrig(ss, y, period = 12, frequencies = 1:3, method = "direct")
model2 <- bsts(y, state.specification = ss, niter = 200)
plot(model2)